If 𝑦=𝑠𝑖𝑛𝑥, determine the fourth derivative of y with respect to xQuestion 4Answera.𝑦𝑖𝑣=𝑠𝑖𝑛𝑥b.𝑦𝑖𝑣=-𝑐𝑜𝑠𝑥c.𝑦𝑖𝑣=𝑐𝑜𝑠𝑥d.𝑦𝑖𝑣=-𝑠𝑖𝑛𝑥

1. Break Down the Problem

We need to find the fourth derivative of the function $y = \sin x$ with respect to $x$.

2. Relevant Concepts

To find the derivatives of $y = \sin x$:

• First derivative: $y' = \frac{dy}{dx}$
• Second derivative: $y'' = \frac{d^2y}{dx^2}$
• Third derivative: $y''' = \frac{d^3y}{dx^3}$
• Fourth derivative: $y^{(4)} = \frac{d^4y}{dx^4}$

3. Analysis and Detail

1. First Derivative: $y' = \frac{d}{dx}(\sin x) = \cos x$

2. Second Derivative: $y'' = \frac{d}{dx}(\cos x) = -\sin x$

3. Third Derivative: $y''' = \frac{d}{dx}(-\sin x) = -\cos x$

4. Fourth Derivative: $y^{(4)} = \frac{d}{dx}(-\cos x) = \sin x$

4. Verify and Summarize

• The first derivative is $\cos x$.
• The second derivative is $-\sin x$.
• The third derivative is $-\cos x$.
• The fourth derivative returns to $\sin x$.

Final Answer

The fourth derivative of $y = \sin x$ with respect to $x$ is: $y^{(4)} = \sin x$

This aligns with option (a), which states that $y^{(4)} = \sin x$.